1. The General (LISREL) SEM model Ulf H. Olsson Professor of statistics

2. Ulf H. Olsson Making Numbers Loyalty Branch Loan Savings Satisfaction

3. Ulf H. Olsson CFA and SEM

4. Ulf H. Olsson CFA and SEM

5. Ulf H. Olsson CFA and SEM No differences in estimation and testing Many estimators ML GLS ULS WLS DWLS

6. Ulf H. Olsson Notation and Background Satorra & Bentler, 1988, equation 4.1

7. Ulf H. Olsson C1 for ML and GLS

8. Ulf H. Olsson The four different chi-squares C1 is N-1 times the minimum value of a fit-function C2 is N-1 times the minimum value of a weighted (involving a weight matrix) fit function under multivariate normality C3 is the Satorra-Bentler Scaled chi-square C4 is N-1 times the minimum value of a weighted (involving a weight matrix) fit function under multivariate non-normality

9. Ulf H. Olsson Asymptotic covariance matrix not provided ULSGLSMLWLSDWLS C10**00 C2***00 C300000 C400000

10. Ulf H. Olsson Asymptotic covariance matrix provided ULSGLSMLWLSDWLS C10***0 C2***0* C3***0* C4***0*

11. Ulf H. Olsson ESTIMATORS If the data are continuous and approximately follow a multivariate Normal distribution, then the Method of Maximum Likelihood is recommended. If the data are continuous and approximately do not follow a multivariate Normal distribution and the sample size is not large, then the Robust Maximum Likelihood Method is recommended. This method will require an estimate of the asymptotic covariance matrix of the sample variances and covariances. If the data are ordinal, categorical or mixed, then the Diagonally Weighted Least Squares (DWLS) method for Polychoric correlation matrices is recommended. This method will require an estimate of the asymptotic covariance matrix of the sample correlations.

12. Ulf H. Olsson Problems with the chi-square test The chi-square tends to be large in large samples if the model does not hold It is based on the assumption that the model holds in the population It is assumed that the observed variables comes from a multivariate normal distribution => The chi-square test might be to strict, since it is based on unreasonable assumptions?!

13. Ulf H. Olsson Alternative test- Testing Close fit

14. Ulf H. Olsson How to Use RMSEA Use the 90% Confidence interval for EA Use The P-value for EA RMSEA as a descriptive Measure RMSEA< 0.05 Good Fit 0.05 < RMSEA < 0.08 Acceptable Fit RMSEA > 0.10 Not Acceptable Fit

15. Ulf H. Olsson Other Fit Indices CN RMR GFI = 1-(Fm/Fn) AGFI= 1 – (k(k+1)/(2df)) (1-GFI) Evaluation of Reliability MI: Modification Indices

16. Ulf H. Olsson Nested Models and parsimony Modification Indices  chi-sq is chi-sq with df=  df Nested Models Re-specification (Modification indices)

17. Ulf H. Olsson RMSEA

18. Ulf H. Olsson RMSEA

19. Ulf H. Olsson LISREL SYNTAX